Tuesday, August 01, 2017

Division by Zero in the 'Mu' Dialectical Arithmetic.

Dear Readers,

See www.dialectics.org :  division by 0 [renamed ‘empty zero’] is readily performed, and yields a finitary, ‘quanto-qualitative’ value [called ‘full zero’], in the axiomatic system of the ‘Mu’ dialectical arithmetic.  This arithmetic also enables a fully arithmetical, fully algorithmic, fully ideographical expression of “dimensional analysis”, instead of the historically retrograde, “syncopated”-algebraic form presently prevalent, e.g., (1)[gm.], (2)[cm.], (3)[sec.], etc.

In this ‘Mu’ dialectical arithmetic, ‘quanto-qualitative’ subscripts/denominators behave analogously to the conventional superscripts extracted by logarithm functions:  script-level multiplications translate to subscript-level, non-amalgamative additions; script-level divisions translate to subscript-level subtractions.

Dynamical system state variable and control parameter ‘arithmetical quantifiers’ can be expressed in forms fully ‘‘‘qualified’’’ by ‘metrical arithmetical qualifier meta-numerals’, and by ‘ontological arithmetical qualifier meta-numerals’, in this ‘Mu’ system of arithmetic.

As a consequence, dynamical “singularities”, arising in finite time, resulting from division by zero in the dynamical differential equation itself, and/or in its solution-function, can be ‘semantified’ -- rendered in their real, Gödelian-like meaning -- in a way which naturally avoids ‘infinity residuals’, i.e., the ‘infinitely erroneous’ values that result from interpreting “singularities” as signifying physical “purely”-quantitative infinities.

Dynamical singularities typically signify a change in the physical ontology that the dynamical equation models -- a change that goes beyond what the language and ‘‘‘ontological commitments’’’ of the model specification, implicit or explicit, can encompass.  That is, they signify an ‘ontological revolution’. 

Thus, the arising of the ‘full zero’ value ‘meta-numeral’ is typically a sign-post that the dynamical trajectory has reached a point where such ontological incompleteness of the model specification becomes explicitly manifest.

‘Arithmetical qualifiers’ are not a new ‘ideo-phenomenon’ in the history of human cognitions regarding numbers, or «arithmoi».  In the prelude to cuneiform in ancient Mesopotamia, ideograms representing both ‘ontological qualifiers’ and ‘metrological qualifiers’, as well as their ‘‘‘quantifiers’’’, eventually emerged, as documented in the work of Dr. Denise Schmandt-Besserat. 

The circa 250 C.E. proto-algebraic text by Diophantus of Alexandria, entitled  “«Arithmetiké»”, which pioneered the movement toward symbolic or ideographical algebra, featured a syncopated proto-symbol, ‘M^o’, short for «Monad», or “unit”, which denoted a qualitative unit of a given kind of object, or, indifferently, of a metrological unit -- i.e., of a “qualitative unit” in either case. 

In Western mathematics after the Renaissance, the presence of ‘arithmetical qualifier meta-numerals’ in mathematical expressions entered a long eclipse and elision, in favor of “purely quantitative” ideographies -- except for the unrecognized presence of ‘metrological unit qualifiers’ in “syncopated” form, e.g., units of measurement like “sec.”, “gm.”, and “cm.”, when expressing “physical quantities”.

Later, an immanent re-emergence of a new kind of ‘arithmetical qualifiers’ ensued, with the discovery that the square root of negative unity could be represented by the so-called “imaginary unit”, i, with rich mathematical consequences, and with the subsequent development of the “hypernumbers” involved in, e.g., the Hamilton Quaternions, the Cayley/Graves Octonions/Octaves, and the “Grassmann [hyper]numbers”, all of which involve “qualitative units”.  “Vector” symbols also represent something beyond the “purely quantitative”, combining “scalar” quantity with ‘directionality’, or orientation.

Likewise, the recent development of set theory, and of the various “orders” of predicate calculus in mathematical logic, all involve the ideographical symbolization of ‘‘‘idea-objects’’’ that are not “pure quantities”, e.g., that represent “qualities”.

starting with --









No comments:

Post a Comment